Analytical methods for non-linear fractional Kolmogorov-Petrovskii-Piskunov equation: Soliton solution and operator solution

Kolmogorov-Petrovskii-Piskunov (KPP) equation can be regarded as a generalized form of the Fitzhugh-Nagumo, Fisher and Huxley equations which have many applications in physics, chemistry and biology. In this paper, two fractional extended versions of the non-linear KPP equation are solved by analytical methods. Firstly, a new and more general fractional derivative is defined and some properties of it are given. Secondly, a solution in the form of operator representation of the non-linear KPP equation with the defined fractional derivative is obtained. Finally, some exact solutions including kink-soliton solution and other solutions of the non-linear KPP equation with Khalil et al.?s fractional derivative and variable coefficeints are obtained. It is shown that the fractional-order affects the propagation velocitie of the obtained kink-soliton solution.

On the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem

We study the compactness property of operator solutions to certain operator inequalities arising from the frequency theorem of Likhtarnikov — Yakubovich for C0-semigroups. We show that the operator solution can be described through solutions of an adjoint problem as it was previously known under some regularity condition. Thus we connect some regularity properties of the semigroup with the compactness of the operator in the general case. We also prove several results useful for checking the non-compactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases arising in applications. As an example, we apply these theorems for a scalar delay equation posed in a proper Hilbert space and show that the operator solution cannot be compact. This results are related to the author recent work on a non-local reduction principle of cocycles (non-autonomous dynamical systems) in Hilbert spaces.

SPECTRUM OF SPIN-½ SYSTEM DRIVEN BY RESONANT EXPONENTIAL PULSE

The exact operator solution for a system of a single 2-level atom (spin-1/2 system) driven by a short resonant exponential pulse is used to examine the atomic dynamics and to calculate the time-dependent spectrum of the fluorescent radiation. The spectrum is expressed analytically in terms of the incomplete Gamma funtion (of complex arguments). Computational display of the results are presented for various system parameters.

GENERALIZED STATISTICS FIELDS AND THE TWO-DIMENSIONAL DERIVATIVE-COUPLING MODEL REVISITED

We reexamine the two-dimensional derivative-coupling model within the operator approach that takes into account the quantum corrections of the fully bosonized Hamiltonian. As a consequence of the fact that there is no true spin-statistics theorem in two dimensions, the operator solution is rewritten in terms of a generalized Mandelstam soliton operator with continuous Lorentz spin (generalized statistics). The particular case corresponding to the Schroer model is considered: the effect of the vector-current-derivative coupling is to give a noncanonical scale dimension for the underlying free Fermi field promoting it to a free Mandelstam operator with generalized statistics.